This Article 
 Bibliographic References 
 Add to: 
Asymmetric Tensor Field Visualization for Surfaces
Dec. 2011 (vol. 17 no. 12)
pp. 1979-1988
Darrel Palke, Oregon State University
Zhongzang Lin, Oregon State University
Guoning Chen, SCI, University of Utah
Harry Yeh, Oregon State University
Paul Vincent, Oregon State University
Robert Laramee, Swansea University
Eugene Zhang, Oregon State University
Asymmetric tensor field visualization can provide important insight into fluid flows and solid deformations. Existing techniques for asymmetric tensor fields focus on the analysis, and simply use evenly-spaced hyperstreamlines on surfaces following eigenvectors and dual-eigenvectors in the tensor field. In this paper, we describe a hybrid visualization technique in which hyperstreamlines and elliptical glyphs are used in real and complex domains, respectively. This enables a more faithful representation of flow behaviors inside complex domains. In addition, we encode tensor magnitude, an important quantity in tensor field analysis, using the density of hyperstreamlines and sizes of glyphs. This allows colors to be used to encode other important tensor quantities. To facilitate quick visual exploration of the data from different viewpoints and at different resolutions, we employ an efficient image-space approach in which hyperstreamlines and glyphs are generated quickly in the image plane. The combination of these techniques leads to an efficient tensor field visualization system for domain scientists. We demonstrate the effectiveness of our visualization technique through applications to complex simulated engine fluid flow and earthquake deformation data. Feedback from domain expert scientists, who are also co-authors, is provided.

[1] P. Alliez, D. Cohen-Steiner,O. Devillers, B. Levy, and M. Desbrun, Anisotropic polygonal remeshing. ACM Transactions on Graphics (SIG-GRAPH 2003), 22 (3): 485–493, July 2003.
[2] W. Benger and H.-C. Hege, Tensor splats. In Visualization and Data Analysis 2004, Proc. of SPIE, volume 5295, pages 151–162, June 2004.
[3] A. Bhalerao and C.-F. Westin, Tensor splats: visualising tensor fields by texture mapped volume rendering. In Sixth International Conference on Medical Image Computing and Computer-Assisted Intervention (MIC-CAI'03), pages 294–901, Montreal, Canada, November 2003.
[4] B. Cabral and L. C. Leedom, imaging vector fields using line integral convolution. In Poceedings of ACM SIGGRAPH 1993, Annual Conference Series, pages 263–272, 1993.
[5] G. Chen, K. Mischaikow, R. S. Laramee, P. Pilarczyk, and E. Zhang, Vector field editing and periodic orbit extraction using Morse decomposition. IEEE Transactions on Visualization and Computer Graphics, 13 (4): 769– 785, 2007.
[6] W. C. de Leeuw and J. J. van Wijk, A probe for local flow field visualization. In Proceedings of the 4th conference on Visualization '93, VIS '93, pages 39–45, Washington, DC, USA, 1993. IEEE Computer Society.
[7] T. Delmarcelle and L. Hesselink, Visualizing second-order tensor fields with hyperstream lines. IEEE Computer Graphics and Applications, 13 (4): 25–33, July 1993.
[8] L. Feng, I. Hotz, B. Hamann, and K. Joy, Anisotropic noise samples. IEEE Transactions on Visualization and Computer Graphics, 14: 342– 354, 2008.
[9] A. Hertzmann and D. Zorin, Illustrating smooth surfaces. Computer Graphics Proceedings, Annual Conference Series (SIGGRAPH 2000), pages 517–526, Aug. 2000.
[10] L. Hesselink, Y. Levy, and Y. Lavin, The topology of symmetric, second-order 3D tensor fields. IEEE Transactions on Visualization and Computer Graphics, 3 (1): 1–11, Mar. 1997.
[11] M. Hlawitschka and G. Scheuermann, HOT lines: tracking lines in higher order tensor fields. In Proceedings IEEE Visualization 2005, pages 27– 34, 2005.
[12] M. Hlawitschka, G. Scheuermann, and B. Hamann, Interactive glyph placement for tensor fields. In Proceedings of the 3rd international conference on Advances in visual computing - Volume Part I, ISVC'07, pages 331–340,Berlin, Heidelberg, 2007. Springer-Verlag.
[13] H. Hotz, L. Feng, H. Hagen, B. Hamann, K. Joy, and B. Jeremic, Physically based methods for tensor field visualization. In Proceedings IEEE Visualization 2004, pages 123–130, 2004.
[14] K. W. Hudnut, Coseismic displacements of the 1992 Landers earthquake sequence. Bull. Seism. Soc. Am., 84: 625–645, 1994.
[15] B. Jobard and W. Lefer, Creating evenly–spaced streamlines of arbitrary density. In Proceedings of the Eurographics Workshop on Visualization in Scientific Computing '97, volume 7, pages 45–55, 1997.
[16] G. Kindlmann, Superquadric tensor glyphs. In Proceedings of IEEE TVCG/EG Symposium on Visualization 2004, pages 147–154, May 2004.
[17] G. Kindlmann and C.-F. Westin, Diffusion tensor visualization with glyph packing. IEEE Transactions on Visualization and Computer Graphics (Proceedings Visualization / Information Visualization 2006), 12 (5): 1329–1335, September-October 2006.
[18] R. M. Kirby, H. Marmanis, and D. H. Laidlaw, Visualizing multivalued data from 2D incompressible flows using concepts from painting. In Proceedings IEEE Visualization '99, pages 333–340.ACM Press, Oct.25–29 1999.
[19] D. H. Laidlaw, E. T. Ahrens, D. Kremers, M. J. Avalos, R. E. Jacobs, and C. Readhead, Visualizing diffusion tensor images of the mouse spinal cord. Visualization Conference, IEEE, pages 127–134, 1998.
[20] R. S. Laramee, C. Garth, H. Doleisch, J. Schneider, H. Hauser, and H. Hagen, Visual analysis and exploration of fluid flow in a cooling jacket. In Proceedings IEEE Visualization 2005, pages 623–630, 2005.
[21] R. S. Laramee, H. Hauser, H. Doleisch, F. H. Post, B. Vrolijk, and D. Weiskopf, The state of the art in flow visualization: dense and texture-based techniques. Computer Graphics Forum, 23 (2): 203–221, June 2004.
[22] R. S. Laramee, H. Hauser, L. Zhao, and F. H. Post, Topology-based flow visualization: the state of the art. In The Topology-Based Methods in Visualization Workshop (TopoInVis 2005), Visualization and Mathematics, pages 1–19, 2007.
[23] R. S. Laramee, J. J. van Wijk, B. Jobard, and H. Hauser, ISA and IBFVS: image space based visualization of flow on surfaces. IEEE Transactions on Visualization and Computer Graphics, 10 (6): 637–648, Nov. 2004.
[24] Z. P. Liu and R. J. Moorhead, II. An advanced evenly-spaced streamline placement algorithm. IEEE Transactions on Visualization and Computer Graphics, 12 (5): 965–972, 2006.
[25] X. Mao, Y. Hatanaka, H. Higashida, and A. Imamiya, Image-guided streamline placement on curvilinear grid surfaces. In Proceedings IEEE Visualization '98, pages 135–142, 1998.
[26] M. Marinov and L. Kobbelt, Direct anisotropic quad-dominant remesh-ing. Computer Graphics and Applications, 12th Pacific Conference on (PG'04), pages 207–216, 2004.
[27] T. McLoughlin, R. S. Laramee, R. Peikert, F. H. Post, and M. Chen, Over two decades of integration-based, geometric flow visualization. Computer Graphics Forum, 29 (6): 1807–1829, 2010.
[28] A. Mebarki, P. Alliez, and O. Devillers, Farthest point seeding for efficient placement of streamlines. In Proceedings IEEE Visualization 2005, pages 479–486. IEEE Computer Society, 2005.
[29] J. Palacios and E. Zhang, Rotational symmetry field design on surfaces. ACM Trans. Graph., 26 (3): 55, 2007.
[30] N. Ray, B. Vallet, W. C. Li, and B. Lévy, N-symmetry direction field design. ACM Trans. Graph., 27 (2): 1–13, 2008.
[31] O. Rosanwo, C. Petz, S. Prohaska, H.-C. Hege, and I. Hotz, Dual streamline seeding. In Proceedings of the 2009 IEEE Pacific Visualization Symposium, PACIFICVIS '09, pages 9–16, Washington, DC, USA, 2009. IEEE Computer Society.
[32] P. V. Sander, J. Snyder, S. J. Gortler, and H. Hoppe, Texture mapping progressive meshes. In SIGGRAPH '01: Proceedings of the 28th annual conference on Computer graphics and interactive techniques, pages 409– 416,New York, NY, USA, 2001. ACM.
[33] T. Schultz and G. L. Kindlmann, Superquadric glyphs for symmetric second-order tensors. IEEE Transactions on Visualization and Computer Graphics, 16: 1595–1604, 2010.
[34] K. Shimada, A. Yamada, and T. Itoh, Anisotropic triangulation of parametric surfaces via close packing of ellipsoids. International Journal of Computational Geometry and Applications, 10: 417–440, 2000.
[35] B. Spencer, R. S. Laramee, G. Chen, and E. Zhang, Evenly-spaced streamlines for surfaces: an image-based approach. Computer Graphics Forum, 28 (6): 1618–1631, 2009.
[36] X. Tricoche, G. Scheuermann, and H. Hagen, Tensor topology tracking: a visualization method for time-dependent 2D symmetric tensor fields. In Computer Graphics Forum 20(3) (Eurographics 2001), pages 461–470, Sept. 2001.
[37] G. Turk and D. Banks, Image-guided streamline placement. In ACM SIGGRAPH 96 Conference Proceedings, pages 453–460, Aug. 1996.
[38] J. J. van Wijk, Image based flow visualization for curved surfaces. In Proceedings IEEE Visualization '03, pages 123–130. IEEE Computer Society, 2003.
[39] V. Verma, D. Kao, and A. Pang, A flow-guided streamline seeding strategy. In Proceedings IEEE Visualization 2000, pages 163–170, 2000.
[40] P. Vincent Application of SAR Interferometry to Low-rate Crustal Deformation Fields. PhD thesis, University of Colorado, 1998.
[41] K. Wu, Z. Liu, S. Zhang, and R. Moorhead, Topology-aware evenly spaced streamline placement. Visualization and Computer Graphics, IEEE Transactions on, 16 (5): 791 –801, sept.-oct. 2010.
[42] E. Zhang, J. Hays, and G. Turk, Interactive tensor field design and visualization on surfaces. IEEE Transactions on Visualization and Computer Graphics, 13 (1): 94–107, 2007.
[43] E. Zhang, H. Yeh, Z. Lin, and R. S. Laramee, Asymmetric tensor analysis for flow visualization. IEEE Transactions on Visualization and Computer Graphics, 15 (1): 106–122, 2009.
[44] X. Zheng and A. Pang, HyperLIC. In Proceedings IEEE Visualization 2003. IEEE Computer Society, 2003.
[45] X. Zheng and A. Pang, Topological lines in 3D tensor fields. In Proceedings IEEE Visualization '04, pages 313–320, 2004.
[46] X. Zheng and A. Pang, 2D asymmetric tensor analysis. IEEE Proceedings on Visualization, pages 3–10, Oct 2005.
[47] X. Zheng, B. Parlett, and A. Pang, Topological structures of 3D tensor fields. In Proceedings IEEE Visualization 2005, pages 551–558, 2005.

Index Terms:
Tensor field visualization, asymmetric tensor fields, vector field visualization, fluid dynamics, solid deformation, earthquakeengineering, glyph packing, hyperstreamline placement, view-dependent visualization.
Darrel Palke, Zhongzang Lin, Guoning Chen, Harry Yeh, Paul Vincent, Robert Laramee, Eugene Zhang, "Asymmetric Tensor Field Visualization for Surfaces," IEEE Transactions on Visualization and Computer Graphics, vol. 17, no. 12, pp. 1979-1988, Dec. 2011, doi:10.1109/TVCG.2011.170
Usage of this product signifies your acceptance of the Terms of Use.